Various classes of block diagrams describe computations that can be performed on application specific computational hardware, such as a computer, microcontroller, FPGA, and custom hardware. Classes of such block diagrams include time-based block diagrams, such as those found within Simulink®, from The MathWorks, Inc. of Natick, Mass., state-based and flow diagrams, such as those found within Stateflow®, from The MathWorks, Inc. of Natick, Mass., and data-flow diagrams. A common characteristic among these various forms of block diagrams is that they define semantics on how to execute the diagram.
Historically, engineers and scientists have utilized time-based block diagram models in numerous scientific areas such as Feedback Control Theory and Signal Processing to study, design, debug, and refine dynamic systems. Dynamic systems, which are characterized by the fact that their behaviors change over time, are representative of many real-world systems. Time-based block diagram modeling has become particularly attractive over the last few years with the advent of software packages, such as Simulink®. Such packages provide sophisticated software platforms with a rich suite of support tools that makes the analysis and design of dynamic systems efficient, methodical, and cost-effective.
A dynamic system (either natural or man-made) is a system whose response at any given time is a function of its input stimuli, its current state, and the current time. Such systems range from simple to highly complex systems. Physical dynamic systems include a falling body, the rotation of the earth, bio-mechanical systems (muscles, joints, etc.), bio-chemical systems (gene expression, protein pathways), weather and climate pattern systems, etc. Examples of man-made or engineered dynamic systems include: a bouncing ball, a spring with a mass tied on an end, automobiles, airplanes, control systems in major appliances, communication networks, audio signal processing, nuclear reactors, a stock market, etc. Professionals from diverse areas such as engineering, science, education, and economics build mathematical models of dynamic systems in order to better understand system behavior as it changes with the progression of time. The mathematical models aid in building “better” systems, where “better” may be defined in terms of a variety of performance measures such as quality, time-to-market, cost, speed, size, power consumption, robustness, etc. The mathematical models also aid in analyzing, debugging and repairing existing systems (be it the human body or the anti-lock braking system in a car). The models may also serve an educational purpose of educating others on the basic principles governing physical systems. The models and results are often used as a scientific communication medium between humans. The term “model-based design” is used to refer to the use of block diagram models in the development, analysis, and validation of dynamic systems.
The various kinds of dynamic systems mentioned above may be described in an explicit or implicit formulation of ODEs. In many applications including simulating physical systems, the dynamic systems may also be described in a linearly-implicit formulation of ODEs. Therefore, it is desired to model and execute dynamic systems described in a linearly-implicit formulation of ODEs within block diagram modeling environments.